Zero-modified distribution of a Geometric variableLet X follow a Geometric distribution with probability of success 0.1, or X ∼ G ( 0.1 ). How do I calculate the probability that the zero-modified distribution of X with f X M ( 0 ) = 0.3 is less than 4?

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Zero-modified distribution of a Geometric variableLet X follow a Geometric distribution with probability of success 0.1, or   X  ∼  G  (  0.1  ). How do I calculate the probability that the zero-modified distribution of X with       f    X    M    (  0  )  =  0.3 is less than 4?

Izabella Fisher · Accepted Answer

Step 1The ordinary geometric distribution Y with parameter 0.1 has   Pr  (  Y  =  k  )  =  (  0.9      )          k      −      1        (  0.1  ) for   k  ≥  1.We assume that by the zero-modified distribution X you mean that   Pr  (  X  =  0  )  =  0.3 and that for for   k  ≥  1, we have                     (1)                    Pr        (        X        =        k        )        =        (        0.7        )        Pr        (        Y        =        k        )        =        (        0.7        )        (        0.9                  )                      k            −            1                          (        0.1        )             for   k  ≥  1.Step 2We want   Pr  (  X  &amp;lt;  4  ). This is   Pr  (  X  =  0  )  +  Pr  (  X  =  1  )  +  Pr  (  X  =  2  )  +  Pr  (  X  =  3  ). The formula for calculating   Pr  (  X  =  k  ) for   k  ≥  1 is given in (1).

Zero-modified distribution of a Geometric variable. Let X follow a Geometric distribution with probability of success 0.1, or X∼G(0.1). How do I calculate the probability that the zero-modified distribution of X with f M/X (0)=0.3 is less than 4?

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